They wouldn't recognize him because unlike the other characters, the new Chekov is actually a different person and wouldn't look like Chekov Prime.

I wasn't joking about that one. I really want to see Anton Yelchin's Chekov detained and questioned by 20th century US officials. Without a clue as to why, preferably. It was so unfair to transport him from the elevator and miss the chance to tease the police for a bit longer. Let's give them a third chance.